When looking at motion in a circle, why do they say that $ r \dot{\theta}$ is transverse velocity when it doesn't look like it is a vector?

Question

In my lecture notes it says that $r \dot{\theta}$ is called the transverse velocity of a particle if it is travelling in a circle. What I don't understand is why this is called a velocity when neither $r$ nor $\dot{\theta}$ are vectors? I thought velocity was a vector quantity which you couldn't get by multiplying two scalars together?

Answer 1

The clue is in the word "transverse" which specifies the direction as perpendicular to the position vector in the direction given by the sign. The magnitude is $r \dot{\theta}$, so both magnitude and direction are given.

The language here is somewhat loose to accommodate the fact that the frame of reference is moving with the particle, so the intuitive ideas we have from Euclidean space and inertial frames of reference have to be modified. The direction of the transverse component depends on where the particle is in space, and there isn't an obvious natural unit vector which captures the idea of "transverse" (you can define one of course, and add it to the expression, to give a true vector).